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<div class="add">

<b>let </b><font color="Maroon">b</font> be   <a href="real_1.html#NM1">Real</a>;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ] <b>means </b><font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K1">REAL</a> ;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(  <a href="real_1.html#NM1">Real</a>) <b>-&gt; </b>   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  = <font color="Maroon">b</font> <a href="real_1.html#K5">-</a> <font color="Maroon">a<sub>1</sub></font>;<br/>
<b>consider </b><font color="Maroon">g</font> being   <a href="partfun1.html#NM1">PartFunc</a> of  <a href="numbers.html#K1">REAL</a> , <a href="numbers.html#K1">REAL</a> <b> such that </b><br/><a NAME="E2:36"/><i><font color="Green">E31</font></i>: 
for <font color="Olive">d</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  holds <br/> ( <font color="Olive">d</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font> iff <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">d</font>] )
 <b>and </b><br/><a NAME="E3:36"/><i><font color="Green">E35</font></i>: 
for <font color="Olive">d</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>   st <font color="Olive">d</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font> holds <br/><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">d</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">d</font>)
 <b>from </b><i><a class="ref" href="partfun2.html#S2">PARTFUN2:sch 2</a>();<br/></i>
<b>take </b>
<font color="Maroon">g</font>
;<br/>

<a NAME="E4:36"/><i><font color="Green">E36</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a>  <a href="numbers.html#K1">REAL</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1"><b>proof </b></a><div class="add">

<a NAME="E1:36_1"/>
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K1">REAL</a>  holds <br/><font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font>
 
<b>by </b><i><a class="ref" href="taylor_1.html#D1" target="_self">Def1</a></i>;<br/>
<a NAME="E2:36_1"/><b>then </b>
 <a href="numbers.html#K1">REAL</a>  <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<b>hence </b><a NAME="E3:36_1"/>
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a>  <a href="numbers.html#K1">REAL</a> 
 <b>by </b><i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:36"/><i><font color="Green">E39</font></i>: 
for <font color="Olive">d</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  holds  <font color="Maroon">g</font> <a href="seq_1.html#K2">.</a> <font color="Olive">d</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="real_1.html#K5">-</a> <font color="Olive">d</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">d</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a> ;<br/>

<a NAME="E1:36_2"/>
<font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">d</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="real_1.html#K5">-</a> <font color="Maroon">d</font>
 
<b>by </b><i>, <a class="txt" href="taylor_1.html#E2:36"><i><font color="Green">E31</font></i></a></i>;<br/>
<b>hence </b><a NAME="E2:36_2"/>
<font color="Maroon">g</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">d</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="real_1.html#K5">-</a> <font color="Maroon">d</font>
 <b>by </b><i><a class="txt" href="taylor_1.html#E2:36"><i><font color="Green">E31</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:36"/><i><font color="Green">E40</font></i>: 
for <font color="Olive">d</font> being  <a href="real_1.html#NM1">Real</a>  st <font color="Olive">d</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K1">REAL</a>  holds <br/><font color="Maroon">g</font> <a href="seq_1.html#K2">.</a> <font color="Olive">d</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="real_1.html#K4">*</a> <font color="Olive">d</font></span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">b</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">d</font> be   <a href="real_1.html#NM1">Real</a>;<br/>

<b>assume </b><a NAME="E1:36_3"/>
<font color="Maroon">d</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K1">REAL</a> 
 ;<br/>

<b>thus </b><font color="Maroon">g</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">d</font> = 
<font color="Maroon">b</font> <a href="real_1.html#K5">-</a> <font color="Maroon">d</font>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">d</font></span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">b</font>

;<br/>


</div><b>end;</b></div>
<a NAME="E7:36"/><i><font color="Green">E41</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font> is  <a href="rcomp_1.html#V3">open</a> <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K1">REAL</a> 
 
<b>by </b><i><a class="txt" href="taylor_1.html#E2:36"><i><font color="Green">E31</font></i></a>, <a class="ref" href="fcont_3.html#T4">FCONT_3:4</a>, <a class="ref" href="subset_1.html#D4">SUBSET_1:def 4</a></i>;<br/>
<a NAME="E8:36"/><b>then </b><i><font color="Green">E42</font></i>: 
( <font color="Maroon">g</font> <a href="fdiff_1.html#R2">is_differentiable_on</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font> &amp; ( for <font color="Olive">x</font> being  <a href="real_1.html#NM1">Real</a>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="fdiff_1.html#K2">`|</a> <span class="p2">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> 1 ) )
 
<b>by </b><i><a class="txt" href="taylor_1.html#E2:36"><i><font color="Green">E31</font></i></a>, , <a class="ref" href="fdiff_1.html#T31">FDIFF_1:31</a></i>;<br/>
<a NAME="E9:36"/>
for <font color="Olive">x</font> being  <a href="real_1.html#NM1">Real</a> holds <br/> ( <font color="Maroon">g</font> <a href="fdiff_1.html#R1">is_differentiable_in</a> <font color="Olive">x</font> &amp;  <a href="fdiff_1.html#K1">diff</a> <font color="Maroon">g</font>,<font color="Olive">x</font> <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> 1 )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">d</font> be   <a href="real_1.html#NM1">Real</a>;<br/>

<b>thus </b><a NAME="E1:36_4"/>
<font color="Maroon">g</font> <a href="fdiff_1.html#R1">is_differentiable_in</a> <font color="Maroon">d</font>
 <b>by </b><i><a class="txt" href="taylor_1.html#E2:36"><i><font color="Green">E31</font></i></a>, , <a class="txt" href="taylor_1.html#E3:36"><i><font color="Green">E35</font></i></a>, <a class="ref" href="fdiff_1.html#T16">FDIFF_1:16</a></i>;<br/>

<b>thus </b> <a href="fdiff_1.html#K1">diff</a> <font color="Maroon">g</font>,<font color="Maroon">d</font> = 
<span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="fdiff_1.html#K2">`|</a> <span class="p2">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">d</font>
<b>by </b><i><a class="txt" href="taylor_1.html#E2:36"><i><font color="Green">E31</font></i></a>, <a class="txt" href="taylor_1.html#E3:36"><i><font color="Green">E35</font></i></a>, <a class="ref" href="fdiff_1.html#D8">FDIFF_1:def 8</a></i>
<br/>.= 
 <a href="real_1.html#K1">-</a> 1
<b>by </b><i><a class="txt" href="taylor_1.html#E2:36"><i><font color="Green">E31</font></i></a>, , , <a class="ref" href="fdiff_1.html#T31">FDIFF_1:31</a></i>
;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E10:36"/>
(  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a>  <a href="numbers.html#K1">REAL</a>  &amp; ( for <font color="Olive">x</font> being  <a href="real_1.html#NM1">Real</a> holds <br/> ( <font color="Maroon">g</font> <a href="seq_1.html#K2">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="real_1.html#K5">-</a> <font color="Olive">x</font> &amp; ( for <font color="Olive">x</font> being  <a href="real_1.html#NM1">Real</a> holds <br/> ( <font color="Maroon">g</font> <a href="fdiff_1.html#R1">is_differentiable_in</a> <font color="Olive">x</font> &amp;  <a href="fdiff_1.html#K1">diff</a> <font color="Maroon">g</font>,<font color="Olive">x</font> <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> 1 ) ) ) ) )
 <b>by </b><i><a class="txt" href="taylor_1.html#E2:36"><i><font color="Green">E31</font></i></a>, </i>;<br/>


</div>
